Hopfield neural lattice models with locally Lipschitz coefficients driven by Lévy noise

Renhai Wang, Hailang Bai, Pengyu Chen and Mirelson M. Freitas

Stochastic Processes and their Applications, 2025, vol. 181, issue C

Abstract: In this article, we study the global-in-time solvability and long-term dynamics of a wide class of infinite-dimensional Hopfield neural models on Zd of infinitely many ODEs with a family of locally Lipschitz coefficients driven by Lévy noise. There are three new features of this stochastic model: (1)The Lévy noise is characterized by two sequence of mutually independent two-sided (including negative initial times) Wiener processes and Poisson random measures; (2)The diffusion coefficients of the Lévy noise are locally Lipschitz associated with an appropriate weight; (3)The connection strength ξi,j between the ith and jth neurons has a finite reciprocal-weighted aggregate efficacy in a weak sense. This Lévy noise driven lattice equation is formulated as an abstract one in an infinite-dimensional weighted Hilbert space ℓϱ2. Both global-in-time well-posedness and long-time dynamics of this abstract stochastic system are investigated under certain conditions. In particular, we show that the long-time dynamics of the stochastic systems can be captured by a weakly compact and weakly attracting mean random attractor in the Bochner space L2(Ω̃,ℓϱ2) over a complete filtered probability space (Ω̃,F̃,{F̃t}t∈R,P). It seems that this is the first time to study the well-posedness and dynamics of lattice Hopfield neural models with locally Lipschitz coefficients driven by Lévy noise even in the autonomous case.

Keywords: Hopfield models; Mean random attractor; Weak attractor; Lévy noise (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1016/j.spa.2024.104559

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